Suppose $\mathcal A$ is $R$-algebra, and $\alpha_1, \alpha_2, \cdots , \alpha_n \in \mathcal A$.

Two-sided ideal of $\mathcal A$ generated by $\alpha_1, \alpha_2, \cdots , \alpha_n$ is

$$\left\{ \beta_1 \alpha_1 + \beta_2 \alpha_2 + \cdots + \beta_n \alpha_n: \beta_1, \beta_2, \cdots, \beta_n \in \mathcal A \right\}$$

I am solving this problem. But to show this set is right ideal of $\mathcal A$, I think condition of commutativity of $\mathcal A$ is necessary.

  • $\begingroup$ What i your question? $\endgroup$ – PrudiiArca 2 days ago
  • $\begingroup$ I am not used to noncommutative algebra, so this is not an answer. I think it would suffice if the $a_i$ commute with any element of $A$, so they are from some sort of center ideal (without taking notes it seems to me like that these elements form an ideal, I am not certain though) $\endgroup$ – PrudiiArca 2 days ago

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